HP F2226A - 48GII Graphic Calculator User Manual page 459

( n
f
k
∑
f ( x )
n 0
i.e.,
The polynomial P (x) is referred to as Taylor's polynomial. The order of the
k
residual is estimated in terms of a small quantity h = x-x
polynomial at a value of x very close to x
where ξ is a number near x = x
estimate of the residual, we provide an estimate of the order of the residual in
reference to h, i.e., we say that R
If h is a small number, say, h<<1, then h
k+1 k
h <<h << ...<< h << 1. Thus, for x close to x
elements in the Taylor polynomial, the smaller the order of the residual.
Functions TAYLR, TAYLR0, and SERIES
Functions TAYLR, TAYLR0, and SERIES are used to generate Taylor
polynomials, as well as Taylor series with residuals. These functions are
available in the CALC/LIMITS&SERIES menu described earlier in this Chapter.
Function TAYLOR0 performs a Maclaurin series expansion, i.e., about X = 0,
of an expression in the default independent variable, VX (typically 'X'). The
expansion uses a 4-th order relative power, i.e., the difference between the
highest and lowest power in the expansion is 4. For example,
)
( x )
∑
n
o ( x x )
o
n !
n k
f ( x ) = P ( x ) + R
k k
. The residual if given by
0
( k + ) 1
f ( ξ )
R ( x )
k
k !
. Since ξ is typically unknown, instead of an
0
(x) has an error of order h
k
k+1
will be typically very small, i.e.,
( n )
f ( x )
n
o ( x x )
,
o
n !
1
( x ).
, i.e., evaluating the
0
k + 1
h
,
n+1
, or R ≈ O(h
, the larger the number of
0
Page 13-24
k+1
).